6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
3:41 minutesProblem 6.2.1a
Textbook Question
Hershey Kisses Based on Data Set 38 “Candies” in Appendix B, weights of the chocolate in Hershey Kisses are normally distributed with a mean of 4.5338 g and a standard deviation of 0.1039 g.
a. What are the values of the mean and standard deviation after converting all weights of Hershey Kisses to z scores using z = (x - μ)/σ ?
b. The original weights are in grams. What are the units of the corresponding z scores?
Verified step by step guidance1
Step 1: Understand the z-score formula. The z-score is calculated using the formula: , where is the individual data point, is the mean, and is the standard deviation.
Step 2: Analyze part (a). When converting all weights to z-scores, the mean of the z-scores will always be 0, and the standard deviation of the z-scores will always be 1. This is because the z-score transformation standardizes the data by centering it around 0 and scaling it by the standard deviation.
Step 3: Address part (b). The original weights are measured in grams. However, z-scores are unitless because they represent the number of standard deviations a data point is from the mean. Units are removed during the z-score calculation.
Step 4: Summarize the results. For part (a), the mean of the z-scores is 0, and the standard deviation is 1. For part (b), the z-scores have no units—they are dimensionless.
Step 5: Reflect on the importance of z-scores. Z-scores are useful for comparing data points from different distributions or for identifying how extreme a data point is relative to its distribution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean (μ) and standard deviation (σ). In this context, the weights of Hershey Kisses follow a normal distribution, which allows for the application of z-scores to standardize the data.
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Z-Score
A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. The z-score indicates how many standard deviations an element is from the mean, allowing for comparison across different datasets.
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Units of Measurement
Units of measurement provide a standard for quantifying physical quantities. In the case of z-scores, they are dimensionless because they represent a standardized value derived from the original data. While the original weights of Hershey Kisses are measured in grams, the z-scores do not have units, as they express relative position rather than absolute measurement.
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